
Note that [1]} bounds terms such as \((\ref {957})\)  deterministically.
Similarly,
\( \phantom{\,.} \mathbf {E}| \varepsilon _k|^2 & \lesssim \mathbf {E}|s_{kk} - 1|^2 + \frac{1}{m^2} + \frac{1}{m^4} \mathbf {E}|\alpha _k^{\prime } (S_{nk} - zI_{n-1})^{-1} \alpha _k - \mathrm {tr}(X_{nk}^{\prime } (S_{nk} - zI_{n-1})^{-1} X_{nk} ) |^2 \nonumber \\& \lesssim \frac{1}{m} + \frac{1}{m^2} \mathbf {E}\mathrm {tr}\, S_{nk} (S_{nk} - zI_{n-1})^{-1} S_{nk} (S_{nk} - \bar{z} I_{n-1})^{-1} \nonumber \\& \lesssim \frac{1}{m} \Big (1 + \frac{1}{m v^2} \Big )  \,.\) 
